A diffusion equation to describe scale–and time–dependent dimensions of turbulent interfaces

Abstract

A new geometrical framework describes the phenomenon of scale-and time-dependent dimensions observed in a great variety of multiscale systems and particularly in the field of turbulence. Based on the notions of scale entropy and scale diffusivity, it leads to a diffusion equation quantifying scale entropy and thus fractal dimension in scale space and in time. For a stationary case and a uniform sink of scale entropy flux, the fractal dimension depends linearly on the scale logarithm. Here, this is experimentally verified in the case of turbulent-flames geometry. Consequences for temporal evolution of scalar passive-turbulent interfaces are investigated and compared with experimental data. Finally, some aspects of dynamics exchange between spatial scales in a turbulent jet are also studied.

On the scale diffusivity of a 2-D liquid atomization process analysis

“…Furthermore, the notion of scale distribution can be applied to characterize systems of any shape. On the other hand, a model has been established to describe the evolution of multi-scale systems whose fractal dimension is scale-and time-dependent [12]. In this model the system is described by the scale entropy and the temporal evolution of the system is modeled by a timedependent diffusion equation.…”

“…This equation introduces the scale diffusivity which characterizes the diffusion of the scale entropy in the scale space. The scale entropy diffusion model has been applied to describe turbulent interfaces and turbulent flames [12,13].…”

Entropic-Skins Geometry to Describe Wall Turbulence Intermittency

“…The last decade showed that the multi-scale structure of turbulent interfaces is indeed much more complicated than simple scale invariance. Let us quote the work by Dimotakis and his collaborators who showed that fractal dimension has a scale-dependent character [17] as well as several studies in the field of turbulent combustion showing that pure fractal behaviour can only be a limit [18,19]. After these precisions, we should indicate that the scale-dependency of fractal dimensions or scaling exponents are not the central point of this study.…”

“…If Np(r) is the minimum number of balls of size needed to cover the set Ωp, the volume is simply given by Vp(r) = Np(r)r d . We then introduce Wp(r) = 1/fp(r) as the number of sets Ωp necessary to fill the space and we define scale-entropy [19] by:…”

Fractal Representation of Exergy

“…In this framework, an elementary optimized configuration is reproduced at every scale, from small to large, to maximize the overall efficiency. Another approach to this question is made by the entropic-skins theory [7][8][9], which provides a new geometrical framework for the study of scale-dependent fractal systems in terms of diffusion of scale-entropy in scale-space.…”